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Tuesday, July 31, 2012

Indian scientist proposes solution to math problem

----Shyam Ranganathan

  • CHENNAI: A mathematics problem with a $1-million prize attached to it and one which has major implications in computer applications, like security and machine intelligence, is claimed to have been solved by an Indian working in the United States.Vinay Deolalikar, a scientist working at HP Labs in California, has proposed a solution to the problem, commonly called by mathematicians as ‘Is P=NP?,' in a paper he has published online. The problem is one of the seven listed by the Clay Mathematics Institute for the Millennium Prize worth $1 million, which will be awarded to the successful solver of each problem.The problem refers to the possible equivalence of two classes of problems. ‘NP' problems refer to those that may take different amounts of time to solve, based on the size of the data. But each solution suggested can be checked easily. An example given on is the solution of jigsaw puzzles — it is easy to verify if a solution is correct but to solve the jigsaw puzzle itself may be very difficult.‘P' problems are those that scale in polynomial time with the size of the data. An example is the problem of alphabetically sorting a list of names. A computer can be programmed to sort the names very fast; and adding many more names will make the task difficult only to an extent and it will still be possible for the computer to handle it.Mr. Deolalikar's solution, a 100-page proof published online, says ‘P' is in fact not the same as ‘NP.' This means computer security systems in their current form may be pretty robust to conventional computer-based attacks, but this also means some problems cannot be solved by simply throwing a lot of computer power at them.The proof has generated a lot of interest among mathematicians, and some have started looking at the proof to see whether it will hold. But as Richard Lipton points out, quoting mathematician Yuri Manin, on his blog “A proof becomes a proof after the social act of accepting it as a proof,” and Mr. Deolalikar has to wait for the paper to be published in a refereed journal after mathematicians vet it.Interestingly, in March this year, another of the problems — the Poincare conjecture — was declared solved by Grigoriy Perelman, but the mathematician refused to accept the $1-million award.

Saturday, July 14, 2012

Reflexive Property A quantity is congruent (equal) to itself.  a = a 
Symmetric Property If a = b, then b = a.
Transitive Property If a = b and b = c, then a = c.
Addition Postulate If equal quantities are added to equal quantities, the sums are equal.
Subtraction Postulate If equal quantities are subtracted from equal quantities, the differences are equal.
Multiplication Postulate If equal quantities are multiplied by equal quantities, the products are equal.  (also Doubles of equal quantities are equal.)
Division Postulate If equal quantities are divided by equal nonzero quantities, the quotients are equal. (also Halves of equal quantities are equal.)
Substitution Postulate A quantity may be substituted for its equal in any expression.
Partition Postulate The whole is equal to the sum of its parts.
Also:  Betweeness of Points:  AB + BC = AC
Angle Addition Postulatem<ABC + m<CBD = m<ABD
Construction Two points determine a straight line.
Construction From a given point on (or not on) a line, one and only one perpendicular can be drawn to the line.

Monday, July 9, 2012

new trends in maths

For those aspiring to pursue mathematics at the undergraduate level from the upcoming academic year starting July, the Delhi University (DU) have a range of options to choose from. About 900 seats in total are being offered across colleges in the DU for undergraduate programmes in mathematics.

According to B K Dass, head of department, mathematics, DU, the subject, in the 21st century has gained immense importance. Since technology — essentially computing — has pervaded all aspects of modern society, the use of mathematics and its popularity, have increased enormously.

"Over the last few years, the number of students opting for mathematics has increased. Financial mathematics, mathematical physics, linear programming and games theory are some of the popular combination programmes amongst students," he says.

Dass believes that a large number of students are interested in game theory since it is an emerging industry with tremendous growth prospects. The subject is popular amongst students as it opens avenues for developing applications. Besides, a number of upcoming game development companies require candidates with strong mathematical skills as they need to work on algorithms to develop applications.

Knowledge of maths is applied in all fields of work, from creating software systems to the field of investigation in areas like brain mapping, in lie-detector machines, fingerprinting, etc. Also, in art, the application of maths has become important for preservation of historic sculptures, paintings and works of ancient artists.

A lot of avenues have opened for mathematics students, opines Rita Malhotra, faculty, mathematics, Kamla Nehru College. "In today's world students are more career-oriented and are interested in developing applications. With more computing and tactical introduced in every field, mathematics is becoming popular amongst students. Besides, having the subject knowledge helps develop decision-making skills of students."

Mathematical finance is a new branch in India with a number of options for maths graduates to get into. Students can make career in game applications, banking, insurance, actuarial sciences, research labs, MNCs or go for an MCA or even management, explains Malhotra.